Trees and Keisler's problem (Q5944051)

The author deals with the following problem, posed by Keisler: Let \({\mathcal M}\), \({\mathcal N}\) be two structures with \({\mathcal M}\prec{\mathcal N}\), \(\phi(x_1,\dots, x_n,y)\) a first-order formula. Does there exist a pair of Skolem functions for \(\phi(x_1,\dots, x_n)\) which preserves elementarity? That means, does there exists \(f: M^n\to M\) and \(g: N^n\to N\) with \[ (M,f)\models\forall x_1\dots x_n[\exists y\phi(x_1,\dots, x_n,y)\to \phi(x_1,\dots, x_n,f(x_1,\dots, x_n))] \] and \((M,f)\prec (N, g)\)? This is answered in the negative taking for \({\mathcal M}\) the ranked binary tree and for \({\mathcal N}\) a rather branchless elementary extension of \(M\) (it was shown by Shelah that such extensions exist).